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Erschienen in:
Reproducing Kernel Method for Fractional Derivative with Non-local and Non-singular Kernel Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

On the Chaotic Pole of Attraction with Nonlocal and Nonsingular Operators in Neurobiology

verfasst von : Emile F. Doungmo Goufo, Abdon Atangana, Melusi Khumalo

Erschienen in: Fractional Derivatives with Mittag-Leffler Kernel

Verlag: Springer International Publishing

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Abstract

Until the neurologists J.L. Hindmarsh and R.M. Rose improved the Hodgkin–Huxley model to provide a better understanding on the diversity of neural response, features like pole of attraction unfolding complex bifurcation for the membrane potential was still a mystery. This work explores the possible existence of chaotic poles of attraction in the dynamics of Hindmarsh–Rose neurons with external current input. Combining with fractional differentiation, the model is generalized with introduction of an additional parameter, the non-integer order of the derivative \(\sigma \) and solved numerically thanks to the Haar Wavelets. Numerical simulations of the membrane potential dynamic show that in the standard case the control parameter is \(\sigma =1,\) the nerve cell’s behavior seems irregular with a pole of attraction generating a limit cycle. This irregularity accentuates as \(\sigma \) decreases (\(\sigma =0.8\) and \(\sigma =0.5\)) with the pole of attraction becoming chaotic.

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Vorheriges Kapitel Existence and Uniqueness Results for a Novel Complex Chaotic Fractional Order System
Nächstes Kapitel Modulating Chaotic Oscillations in Autocatalytic Reaction Networks Using Atangana–Baleanu Operator
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Metadaten
Titel
On the Chaotic Pole of Attraction with Nonlocal and Nonsingular Operators in Neurobiology
verfasst von
Emile F. Doungmo Goufo
Abdon Atangana
Melusi Khumalo
Copyright-Jahr
2019
Buch
Fractional Derivatives with Mittag-Leffler Kernel

Print ISBN: 978-3-030-11661-3

Electronic ISBN: 978-3-030-11662-0

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-11662-0_8

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